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Digital Signal Processing

Digital Signal Processing. Prof. Nizamettin AYDIN naydin @ yildiz .edu.tr http:// www . yildiz .edu.tr/~naydin. Digital Signal Processing. Lecture 4 Spectrum Representation. READING ASSIGNMENTS. This Lecture: Chapter 3, Section 3-1 Other Reading: Appendix A: Complex Numbers

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Digital Signal Processing

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  1. Digital Signal Processing Prof. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin

  2. Digital Signal Processing Lecture 4 Spectrum Representation

  3. READING ASSIGNMENTS • This Lecture: • Chapter 3, Section 3-1 • Other Reading: • Appendix A: Complex Numbers • Next Lecture: Ch 3, Sects 3-2, 3-3, 3-7 & 3-8

  4. LECTURE OBJECTIVES • Sinusoids with DIFFERENT Frequencies • SYNTHESIZE by Adding Sinusoids • SPECTRUM Representation • Graphical Form shows DIFFERENT Freqs

  5. FREQUENCY DIAGRAM –250 –100 0 100 250 • Plot Complex Amplitude vs. Freq f (in Hz)

  6. Another FREQ. Diagram A-440 Frequency is the vertical axis Time is the horizontal axis

  7. MOTIVATION • Synthesize Complicated Signals • Musical Notes • Piano uses 3 strings for many notes • Chords: play several notes simultaneously • Human Speech • Vowels have dominant frequencies • Application: computer generated speech • Can all signals be generated this way? • Sum of sinusoids?

  8. Fur Elise WAVEFORM Beat Notes

  9. Speech Signal: BAT • Nearly Periodic in Vowel Region • Period is (Approximately) T = 0.0065 sec

  10. Euler’s Formula Reversed • Solve for cosine (or sine)

  11. INVERSE Euler’s Formula • Solve for cosine (or sine)

  12. SPECTRUM Interpretation • One has a positive frequency • The other has negative freq. • Amplitude of each is half as big • Cosine = sum of 2 complex exponentials:

  13. NEGATIVE FREQUENCY • Is negative frequency real? • Doppler Radar provides an example • Police radar measures speed by using the Doppler shift principle • Let’s assume 400Hz 60 mph • +400Hz means towards the radar • -400Hz means away (opposite direction) • Think of a train whistle

  14. SPECTRUM of SINE • Sine = sum of 2 complex exponentials: • Positive freq. has phase = -0.5p • Negative freq. has phase = +0.5p

  15. GRAPHICAL SPECTRUM w -7 0 7 EXAMPLE of SINE AMPLITUDE, PHASE & FREQUENCY are shown

  16. SPECTRUM ---> SINUSOID –250 –100 0 100 250 f (in Hz) • Add the spectrum components: What is the formula for the signal x(t)?

  17. Gather (A,w,f) information Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz Amplitude & Phase 4 -p/2 7 +p/3 10 0 7 -p/3 4 +p/2 Note the conjugate phase DC is another name for zero-freq component DC component always has f=0 or p (for real x(t) )

  18. Add Spectrum Components-1 Amplitude & Phase 4 -p/2 7 +p/3 10 0 7 -p/3 4 +p/2 Frequencies: -250 Hz -100 Hz 0 Hz 100 Hz 250 Hz

  19. Add Spectrum Components-2 –250 –100 0 100 250 f (in Hz)

  20. Simplify Components Use Euler’s Formula to get REAL sinusoids:

  21. FINAL ANSWER So, we get the general form:

  22. Summary: GENERAL FORM

  23. Example: Synthetic Vowel • Sum of 5 Frequency Components

  24. SPECTRUM of VOWEL • Note: Spectrum has 0.5Xk(except XDC) • Conjugates in negative frequency

  25. SPECTRUM of VOWEL (Polar Format) 0.5Ak fk

  26. Vowel Wavefor (sum of all 5 components)

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